Kinetic-energy threshold for Cherenkov radiation ($K_{\mathrm{th}}=m{c}^2[(1-n^{-2}{)}^{-1/2}-1]$)

At a specified frequency with transparent-medium phase index $n>1$, Cherenkov radiation requires $\beta >1/n$. Inserting the threshold speed into relativistic kinetic energy gives $K_{\mathrm{th}}=m{c}^2[(1-n^{-2}{)}^{-1/2}-1]$. Equality is the kinematic limit; the cone angle and emission vanish there. A dispersive material has different thresholds in different spectral intervals. The result uses phase velocity, not group velocity.